Group #26130637 — subgroups

Subgroups — #26130637

Order 12032.

204 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245663	C₂	1	1	yes	maximal, characteristic
2	#20242782	C₂	1	2	yes	maximal
3	#20241460	C₂	1	1	yes	maximal, characteristic
4	#20241442	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15689528	C₂²	1	2	yes	—
7	#15689333	—	2	2	no	—
8	#15688949	—	2	1	no	—
9	#15688949	—	2	1	no	—
10	#15688876	C₂²	1	2	yes	—
11	#15688868	—	2	2	no	—
12	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×C₄⋊C₁₆	—	2	1	no	—
14	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₄₇	—	2	1	no	—
16	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#10625337	—	4	2	no	—
18	#13094024	—	2	2	no	—
19	#13093958	—	2	2	no	—
20	#13093958	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	#13093947	—	4	2	no	—
23	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₄₇	—	2	1	no	—
25	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₄₇	—	2	1	no	—
28	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₄×C₈×C₄₇	—	2	1	no	—
30	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×Q₈×C₄₇	D₈	1	1	yes	characteristic
32	C₂×Dic₉₄	—	4	2	no	—
33	C₂×C₈×C₄₇	—	2	1	no	—
34	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
36	C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
38	C₄²×C₄₇	—	2	1	no	—
39	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
40	#10625337	—	8	4	no	—
41	C₄⋊Dic₄₇	—	4	2	no	—
42	C₄⋊Dic₄₇	—	4	2	no	—
43	C₂×C₄₇⋊C₈	—	4	2	no	—
44	C₁₆×C₄₇	—	2	1	no	—
45	C₁₆×C₄₇	—	4	1	no	—
46	C₁₆×C₄₇	Q₁₆	1	2	yes	complemented
47	Q₈×C₄₇	—	2	1	no	—
48	Q₈×C₄₇	D₁₆	1	2	yes	—
49	C₂×C₄×C₄₇	—	2	1	no	—
50	C₂×C₄×C₄₇	—	2	1	no	—
51	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
54	C₂×Dic₄₇	—	8	2	no	—
55	Dic₉₄	—	8	2	no	—
56	Dic₉₄	—	8	2	no	—
57	C₈×C₄₇	—	2	1	no	—
58	C₈×C₄₇	—	4	1	no	—
59	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
60	C₈×C₄₇	#137	1	1	yes	characteristic
61	C₄₇⋊C₈	—	8	2	no	—
62	#12457	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	#466	1	1	yes	characteristic
64	C₄×C₄₇	—	2	2	no	—
65	C₄×C₄₇	—	2	1	no	—
66	C₄×C₄₇	—	4	1	no	—
67	C₄×C₄₇	#462	1	1	yes	characteristic
68	C₄×C₄₇	#473	1	1	yes	characteristic
69	C₄×C₄₇	C₂×D₁₆	1	1	yes	characteristic
70	C₄×C₄₇	#508	1	1	yes	characteristic
71	Dic₄₇	—	16	2	no	—
72	#2252	—	47	1	no	—
73	Q₈×C₁₆	D₄₇	1	1	yes	characteristic
74	#1674	—	47	2	no	—
75	#1371	—	47	1	no	—
76	#1364	—	47	2	no	—
77	C₂×C₄₇	#1674	1	1	yes	characteristic
78	C₂×C₄₇	#2215	1	1	yes	characteristic
79	C₂×C₄₇	#2222	1	1	yes	characteristic
80	#500	—	47	2	no	—
81	#462	—	94	2	no	—
82	C₈×Q₈	C₂×D₄₇	1	1	yes	characteristic
83	#366	—	94	1	no	—
84	#366	—	94	1	no	—
85	C₄⋊C₁₆	—	2	1	no	—
86	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
87	C₄×C₁₆	—	2	1	no	—
88	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
89	#335	—	47	2	no	—
90	#332	—	94	2	no	—
91	C₄₇	#12457	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	—	188	2	no	—
93	C₄⋊Q₈	—	94	2	no	—
94	C₄×Q₈	D₁₈₈	1	1	yes	characteristic
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
97	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
98	#107	—	94	2	no	—
99	#107	—	94	2	no	—
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	—	94	2	no	—
102	C₄⋊C₈	D₁₈₈	1	1	yes	characteristic
103	#103	—	188	2	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
106	C₂×Q₈	—	188	2	no	—
107	C₂×Q₈	D₃₇₆	1	1	yes	characteristic
108	Q₁₆	—	376	4	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	188	2	no	—
111	C₂×C₈	#10625350	1	1	yes	characteristic
112	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	188	2	no	—
115	C₄⋊C₄	—	188	2	no	—
116	C₄⋊C₄	D₃₇₆	1	1	yes	characteristic
117	C₄²	—	2	1	no	—
118	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	4	1	no	—
121	C₁₆	#10625337	1	2	yes	complemented
122	Q₈	—	2	1	no	—
123	Q₈	—	376	2	no	—
124	Q₈	—	376	2	no	—
125	Q₈	D₇₅₂	1	2	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	376	2	no	—
129	C₂×C₄	#13093998	1	1	yes	characteristic
130	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
131	C₂×C₄	#13094046	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	376	2	no	—
135	C₈	C₂×#10625337	1	1	yes	characteristic
136	C₈	#13094054	1	1	yes	characteristic
137	C₂²	#15689359	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	2	no	—
140	C₄	—	4	1	no	—
141	C₄	—	752	2	no	—
142	C₄	#15689333	1	1	yes	characteristic
143	C₄	#15689398	1	1	yes	characteristic
144	C₄	C₂×D₇₅₂	1	1	yes	characteristic
145	C₄	#15689555	1	1	yes	characteristic
146	C₂	#20242782	1	1	yes	characteristic
147	C₂	#20245474	1	1	yes	characteristic
148	C₂	#20245513	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245663

C₂

yes

maximal, characteristic

#20242782

C₂

yes

maximal

#20241460

C₂

yes

maximal, characteristic

#20241442

C₂

yes

maximal

Q₈×C₁₆×C₄₇

C₂

yes

maximal, characteristic

#15689528

C₂²

yes

—

—

—

—

C₂²

yes

—

#15688868

—

C₈×Q₈×C₄₇

C₂²

yes

characteristic

C₄₇×C₄⋊C₁₆

—

C₄₇×C₄⋊C₁₆

C₂²

yes

characteristic

C₄×C₁₆×C₄₇

—

C₄×C₁₆×C₄₇

C₂²

yes

—

—

—

—

C₄⋊(C₄₇⋊C₈)

—

#13093947

—

C₄×Q₈×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

D₈

yes

characteristic

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₈

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

#10625337

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

—

C₁₆×C₄₇

Q₁₆

yes

complemented

Q₈×C₄₇

—

Q₈×C₄₇

D₁₆

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

#135

yes

characteristic

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×Q₁₆

yes

characteristic

C₈×C₄₇

#137

yes

characteristic

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₂×D₁₆

yes

characteristic

C₄×C₄₇

#508

yes

characteristic

Dic₄₇

—

#2252

—

Q₈×C₁₆

D₄₇

yes

—

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

C₈×Q₈

C₂×D₄₇

yes

characteristic

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₄₇

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

#335

—

#332

—

C₄₇

#12457

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

188

—

C₄⋊Q₈

—

C₄×Q₈

D₁₈₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₄₇

yes

characteristic

C₂×C₁₆

C₂²⋊D₄₇

yes

characteristic

#107

—

#107

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

D₁₈₈

yes

characteristic

103

#103

—

188

—

104

C₄×C₈

—

105

C₄×C₈

C₂²×D₄₇

yes

characteristic

106

C₂×Q₈

—

188

—

107

C₂×Q₈

D₃₇₆

yes

characteristic

108

Q₁₆

—

376

—

109

C₂×C₈

—

110

C₂×C₈

—

188

—

111

C₂×C₈

#10625350

yes

characteristic

112

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

188

—

115

C₄⋊C₄

—

188

—

116

C₄⋊C₄

D₃₇₆

yes

characteristic

117

C₄²

—

118

C₄²

C₂×D₁₈₈

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

#10625337

yes

complemented

122

Q₈

—

123

Q₈

—

376

—

124

Q₈

—

376

—

125

Q₈

D₇₅₂

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

376

—

129

C₂×C₄

#13093998

yes

characteristic

130

C₂×C₄

C₂×D₃₇₆

yes

characteristic

131

C₂×C₄

#13094046

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

376

—

135

C₈

C₂×#10625337

yes

characteristic

136

C₈

#13094054

yes

characteristic

137

C₂²

#15689359

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

752

—

142

C₄

#15689333

yes

characteristic

143

C₄

#15689398

yes

characteristic

144

C₄

C₂×D₇₅₂

yes

characteristic

145

C₄

#15689555

yes

characteristic

146

C₂

#20242782

yes

characteristic

147

C₂

#20245474

yes

characteristic

148

C₂

#20245513

yes

characteristic

	subgroup	G/H
center	C₂²	#15689359
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	Q₈×C₁₆×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	#466
hypercenter	Q₈×C₁₆	D₄₇
hypocenter	C₄₇	#12457
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#15689359

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

Q₈×C₁₆×C₄₇

C₂

socle, prime radical

C₂²×C₄₇

#466

hypercenter

Q₈×C₁₆

D₄₇

hypocenter

C₄₇

#12457

LCS

G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
46. C₁₆×C₄₇	Q₁₆	semidirect
91. C₄₇	#12457	semidirect
121. C₁₆	#10625337	semidirect

H ≅ G/N

type

46. C₁₆×C₄₇

Q₁₆

semidirect

91. C₄₇

#12457

semidirect

121. C₁₆

#10625337

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20245663	2
#20242782	2
#20241460	2
#20241442	2
Q₈×C₁₆×C₄₇	2
#12457	47

subgroup

index

Q₈×C₁₆×C₄₇

Subgroups — #26130637

Conjugacy classes of subgroups

Complements

Maximal subgroups